Let ρ: R→R be a continuous, strictly increasing function (so y>x ⇒ ρ(y) > ρ(x)). Show that d(x,y) = │ρ(y) – ρ(x)│ is a distance function on R, which is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.

here i have to show - d is a distance function- d is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.is that right?

and what does it mean by 'distance function is complete'?

April 4th 2009, 04:45 AM

Plato

Quote:

Originally Posted by jin_nzzang

What does it mean by 'distance function is complete'?

A complete metric is a metric in which every Cauchy sequence is convergent.